Posted by Claire on Tuesday, February 15, 2011 at 3:01am.
Take the example of a differential equation:
Since the differential equation in question has not been posted, I will work with an example.
y"-2y'+2y=e^{2x}.....(1)
The solution yh to the homogeneous equation
y"-2y'+2y=0 .... (2)
is
yh=C1*e^{x}+C2*x*e^{x}
Note the two integration constants.
To obtain the general solution to equation (1), we must add to yh, solution of (2) a particular solution that satisfies (1). This is called a particular solution, yp.
For this particular problem,
yp=x²e^{x}
which can be obtained by the method of variation of parameters.
Therefore the general solution of (1) is the sum of yh and yp, i.e.
y = C1*e^{x} + C2*x*e^{x} + x²e^{x} ...(general solution)
However, y still contains the constants C1 and C2 to be determined. They can be found by the initial conditions, such as y(0)=0 and y'(0)=1.
Substitute (0,0) into the general solution, we obtain an equation in terms of C1 and C2.
Differentiate the general solution once and substitute x=0, y'(0)=1, we get another equation. From these equations, we can solve for C1 and C2 which is the solution to the problem.
The above general solution is explicit because y, and only y, appears only on the left hand side.
A solution of the form
xy²+x²=2y²
is called an implicit solution because y appears more than once in an equation.